# Show compactness of an evolution operator

Consider the heat equation

$$u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$

with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$

and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$.

1.) Show that $u_0\star G_t$ with $G_t(x)=\frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$ is a solution of the heath equation problem above.

2.) Show, that $E(T)\colon L^2([0,1])\to L^2([0,1])$ is a continuous and compact operator.

I already showed 1.) by solving the partial differential equation with Fourier transformation. So far so good.

But I do not come along with the compactness in 2.).

(I showed the continuity with the Young inequation using the convolution - is that right?)

But how can I show the compactness of that operator?

Ok for the continuity. For the compactness, the standard technique is showing that $E(T)$ is continuous as an $L^2([0,1])\to H^1([0,1])$ operator. The claim will follow from the compact imbedding of $H^1([0,1])$ into $L^2([0,1])$. – Giuseppe Negro Jan 28 '13 at 2:45